Axiomatic cellular boundaries are integral incidence matrices with coefficients

Given: The objects and hypotheses in the statement above.

For a CW pair $(X,A)$ and an ordinary theory $h$ with coefficient $G$, put $F_{-1}=A$ and $F_r=A\cup X^r$ for $r\ge 0$. Set $C_r^h(X,A)=h_r(F_r,F_{r-1})(r\ge 0),C_r^h=0(r<0)\mathrm{.}$ Each $C_r^h$ is the direct sum of copies of $G$ indexed by the relative $r$-cells. Define $d_0=0$ and for $r\ge 1$ let $d_r$ be the triple boundary to $h_{r-1}(F_{r-1},A)$ followed by its map to $h_{r-1}(F_{r-1},F_{r-2})$. Then $d_{r-1}{d}_r=0$, naturally for cellular maps of CW pairs. (Any ordinary homology theory has a cellular chain complex on a cw pair)

Let $n\ge 0$ and $u:S^n\to S^n$ be continuous. If $u_{\ast }$ on ${\tilde{H}}_n(S^n;\mathbb{Z})$ is multiplication by $d$, then $u_{\ast }$ on ${\tilde{h}}_n(S^n)\cong G$ for every ordinary theory $h$ is $d{\mathrm{i}\mathrm{d}}_G$. The identifications use the same oriented sphere generator; for $n=0$ use the difference of the two point classes. (Sphere endomorphisms act by the same integer in every ordinary theory)

For $n\ge 2$ and oriented cells $e_{\alpha }^n$ and $e_{\beta }^{n-1}$, collapse the complement of $e_{\beta }^{n-1}$ in $X^{n-1}$ and compose the attaching map of $e_{\alpha }^n$ with the resulting quotient to $S^{n-1}$. Its induced endomorphism of oriented ${\tilde{H}}_{n-1}(S^{n-1};\mathbb{Z})$ is multiplication by a unique integer, denoted $[e_{\alpha }^n:e_{\beta }^{n-1}]$. For $n=1$, orient the characteristic interval of $e_{\alpha }^1$ from $-1$ to $+1$ and, for a vertex $v=e_{\beta }^0$, set $[e_{\alpha }^1:v]={\mathbf{1}}_{\{{\chi }_{\alpha }(+1)=v\}}-{\mathbf{1}}_{\{{\chi }_{\alpha }(-1)=v\}}\mathrm{.}$ Thus an oriented edge contributes its terminal vertex minus its initial vertex, and a loop with both endpoints at one vertex has incidence number zero there. (Incidence number of two CW cells)

Let $X$ be a CW complex. For $n\ge 1$, in the integral cellular chain groups with the chosen cell orientations, $d_n{e}_{\alpha }^n={\sum }_{\beta }[e_{\alpha }^n:e_{\beta }^{n-1}]e_{\beta }^{n-1}\mathrm{.}$ (Cellular boundary is the incidence degree matrix)